P is a proper subset of NP

TL;DR

This paper argues that P cannot equal NP by analyzing the complexity of solving NP-complete problems and demonstrating that polynomial-time solutions cannot process all necessary input sets, concluding P is a proper subset of NP.

Contribution

It provides a proof that P does not equal NP by examining the growth of input sets and complexity classes involved in solving NP-complete problems.

Findings

The number of input sets approaches infinity as clauses increase in NP-complete problems.

Polynomial-time solutions cannot process more than a polynomial number of input sets.

Subdividing input sets into polynomial search partitions is in the FEXP complexity class.

Abstract

The purpose of this article is to examine and limit the conditions in which the P complexity class could be equivalent to the NP complexity class. Proof is provided by demonstrating that as the number of clauses in a NP-complete problem approaches infinity, the number of input sets processed per computation performed also approaches infinity when solved by a polynomial time solution. It is then possible to determine that the only deterministic optimization of a NP-complete problem that could prove P = NP would be one that examines no more than a polynomial number of input sets for a given problem. It is then shown that subdividing the set of all possible input sets into a representative polynomial search partition is a problem in the FEXP complexity class. The findings of this article are combined with the findings of other articles in this series of 4 articles. The final conclusion will be demonstrated that P =/= NP.